Suppose that $\{\mu_k\}$ is a sequence of probability measures defined on $(I,\mathcal{B})$, where $I$ is a closed real interval and $\mathcal{B}$ is its Borel $\sigma$-algebra, such that:

- each $\mu_k$ is made of countably many atoms $\{p^n_k\}$;
- for every $k$, the set $\bigcup_{n\in\mathbb{N}}p^n_k$ is not dense in $I$;
- $\bigcup_{n,k\in\mathbb{N}}p^n_k$ is dense in $I$.

I wonder which further assumptions are needed to ensure that:

- $\mu_k$ weakly converges to a probability measure $\mu$,
- $\mu$ is absolutely continuous.

Thanks a lot!

uniform absolute continuity. $\endgroup$ – Nate Eldredge Nov 15 '20 at 19:06